28 History of the Theory of Numbers. [Chap, i 
and considered it probable that primes of the forms 2''±1, 2'^S (if not 
yielding Lucassians) generally yield prime values of 2^ — 1, and that no 
other primes will. All known and conjectured primes 2^ — 1, with p prime, 
fall under this rule. 
In a letter to Tannery/^- Lucas stated that Mersenne^°'^^ implied that 
a necessary and sufficient condition that 2^—1 be a prime is that p be a 
prime of one of the forms 2^"+l, 2^''±3, 2^"+^ — 1. Tannery expressed his 
behef that the theorem was empirical and due to Frenicle, rather than to 
Fermat, and noted that the sufficient condition would be false if 2^' — 1 is 
composite [as is the case, Fauquembergue^^°]. 
Goulard and Tannery^^^ made minor remarks on the subject of the last 
two papers. 
A. Cunningham^^ found that 2^^' - 1 has the factor 7487. This contra- 
dicts LeLasseur's^^- statement on di visors < 30000 of Mersenne's numbers. 
A. Cunningham^^^ found 13 new cases (317, 337, 547, 937, . . .) in which 
2^—1 is composite, and stated that for the 22 outstanding primes 5^257 
[above list^^- except 61, 197] 2^ — 1 has no divisor < 50,000 (error as to 
q = lSl, see Woodall^^). The factors obtained in the mentioned 13 cases 
were found after much labor by the indirect method of Bickmore,^^^ who 
gave the factors 1913 and 5737 of 2-^^-1. 
A. Cunningham^" gave a factor of 2^-1 for g = 397, 1801, 1367, 5011 
and for five larger primes q. 
C. Bourlet"^ proved that the sum of the reciprocals of all the divisors 
di of a perfect number n equals 2 [Catalan ^^^], by noting that n/di ranges 
with di over the divisors of n, so that 2n = 'Zn/di. The same proof occurs 
in II Pitagora, Palermo, 16, 1909-10, 6-7. 
M. Stuyvaert^^^ remarked that an odd perfect number, if it exists, is a 
sum of two squares since it is of the form pk^, where p is a prime 4n+l 
[Frenicle,^ Euler^sj 
T. Pepin^"° proved that an odd perfect number relatively prime to 3-7, 
3-5 or 3-5-7 contains at least 11, 14 or 19 distinct prime factors, respectively, 
and can not have the form 6/cH-5. 
F. J. Studnicka^^^ called Ep = 2''-\2''-l) an Euclidean number if 2^-1 
is a prime. The product of all the divisors <Ep of Ep is E/~'^. When 
Ep is written in the diadic system (base 2), it has 2p — l digits, the first p of 
which are unity and the last p — 1 are zero. 
"^L'interm^diaire des math., 2, 1895, 317. 
i«/6Mi., 3, 1896, 115, 188, 281. 
^"Nature, 51, 1894-5, 533; Proc. Lond. Math. Soc, 26, 1895, 261; Math. Quest. Educat. Times, 
5, 1904, 108, last footnote. 
i«British Assoc. Reports, 1895, 614. 
i«On the numerical factors of a"-l, Messenger Math., 25, 1895-6, 1-44; 26, 1896-7, 1-38. 
French transl. by Fitz-Patrick, Sphinx-Oedipe, 1912, 129-144. 155-160. 
"Troc. London Math. Soc, 27, 1895-6, 111. 
"8Nouv. Ann. Math., (3), 15, 1896, 299. 
"•Mathesis, (2), 6, 1896, 132. 
''"Memou-e Accad. Pont. Nuovi Lincei, 13, 1897, 345-420. 
"'Sitzungsber. Bohm. Gesell., Prag, 1899, math, nat., No. 30. 
